chap3.tex 23.7 KB
 Olivier committed Mar 23, 2017 1 \renewcommand{\lastedityear}{2017}  Olivier committed Sep 23, 2017 2  \renewcommand{\lasteditmonth}{09}  Olivier committed Mar 27, 2019 3  \renewcommand{\lasteditday}{27}  Olivier committed Dec 12, 2015 4 \renewcommand{\numberofthischapter}{3}  Olivier committed Mar 29, 2019 5 \renewcommand{\titleofthischapter}{\namechapterthreetitle}  Olivier committed Apr 03, 2015 6   Olivier committed Dec 12, 2015 7 \fluidmechchaptertitle  Olivier committed Mar 19, 2019 8 \label{chap_three}  Olivier committed Apr 03, 2015 9 10  \mecafluboxen  Olivier committed Mar 18, 2019 11 \mecafluboxtmp  Olivier committed Apr 03, 2015 12   Olivier committed Mar 24, 2016 13 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  Olivier committed Apr 03, 2015 14 15 \section{Motivation}  Olivier committed Sep 23, 2017 16  \youtubethumb{1LXlFVtPoCY}{pre-lecture briefing for this chapter, part 1/2}{\oc (\ccby)}  Olivier committed Apr 03, 2015 17 18  Our objective for this chapter is to answer the question “what is the \emph{net} effect of a given fluid flow through a given volume?”.  Olivier committed Apr 25, 2016 19  Here, we develop a mass, momentum and energy accounting methodology to analyze the flow of continuous medium. This method is not powerful enough to allow us to describe extensively the nature of fluid flow around bodies; nevertheless, it is extremely useful to quantify forces, moments, and energy transfers associated with fluid flow.  Olivier committed Apr 03, 2015 20   Olivier committed Mar 24, 2016 21 22  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  Olivier committed Apr 03, 2015 23 24 25 26 \section{The Reynolds transport theorem} \subsection{Control volume}  Olivier committed Apr 23, 2017 27  Let us begin by describing the flow which interests us a a generic velocity field $\vec V = (u, v, w)$ which is a function of space and time ($\vec V = f(x, y, z, t)$).  Olivier committed Apr 03, 2015 28   Olivier committed Apr 23, 2017 29  Within this flow, we are interested in an arbitrary volume named \vocab{control volume} (CV) which is free to move and change shape (\cref{fig_cv}). We are going to measure the properties of the fluid at the borders of this volume, which we call the \vocab{control surface} (CS), in order to compute the net effect of the flow through the volume.  Olivier committed Apr 03, 2015 30 31 32  \begin{figure} \begin{center}  Olivier committed Mar 24, 2016 33  \includegraphics[width=0.6\textwidth]{concept_control_volume_system.png}  Olivier committed Apr 03, 2015 34  \end{center}  Olivier committed Mar 29, 2019 35  \supercaption{A control volume within a flow. The \vocab{system} is the amount of mass included within the control volume at a given time. At a later time, it may have left the control volume, and its shape and properties may have changed. The control volume may also change shape with time, although this is not represented here.}{\wcfile{System control volume integral analysis.svg}{Figure} \cczero \oc}  Olivier committed Apr 20, 2015 36  \label{fig_cv}  Olivier committed Apr 03, 2015 37 38  \end{figure}  Olivier committed Apr 23, 2017 39  At a given time, the control volume contains a certain amount of mass which we call the \vocab{system} (sys). Thus the system is a fixed amount of mass transiting through the control volume at the time of our study, and its properties (volume, pressure, velocity etc.) may change in the process.  Olivier committed Apr 25, 2016 40   Olivier committed Apr 23, 2017 41  All along the chapter, we are focusing on the question: based on measured fluid properties at some point in space and time (the properties at the control surface), how can we quantify what is happening to the system (the mass inside the control volume)?  Olivier committed Apr 03, 2015 42 43 44 45  \subsection{Rate of change of an additive property}  Olivier committed Apr 25, 2016 46 47 48  In order to proceed with our calculations, we need a robust accounting methodology. We start with a “dummy” fluid property $B$, which we will later replace with physical variables of interest.\\ Let us thus consider an arbitrary additive property $B$ of the fluid. By the term \vocab{additive} property, we mean that the total \emph{amount of property} is divided if the fluid is divided. For instance, this is true of mass, volume, energy, entropy, but not pressure or temperature.\\ The \vocab{specific} (i.e. per unit mass) value of $B$ is designated $b \equiv B/m$.  Olivier committed Apr 03, 2015 49   Olivier committed Apr 23, 2017 50 51 52 53 54 55  We now wish to compute the variation of a system’s property $B$ based on measurements made at the borders of the control volume. We will achieve this with an equation containing three terms: \begin{itemize} \item The time variation of the quantity $B$ within the system is measured with the term $\timederivative{B_\sys}$. This may represent, for example, the rate of change of the fluid’s internal energy as it travels through a jet engine. \item Within the control volume, the enclosed quantity $B_\cv$ can vary by accumulation (for example, mass may be increasing in an air tank fed with compressed air): we measure this with the term $\timederivative{B_\cv}$. \item Finally, a mass flux may be flowing through the boundaries of the control volume, carrying with it some amount of $B$ every second: we name that net flow out of the system $\dot B_\net \equiv \dot B_\out - \dot B_\inn$. \end{itemize}  Olivier committed Apr 03, 2015 56 57 58  We can now link these three terms with the simple equation: \begin{IEEEeqnarray}{CCCCC}  Olivier committed Apr 20, 2015 59  \timederivative{B_\sys} & = & \timederivative{B_\cv} & + & \dot B_\net \label{eq_rtt_basic}\\\nonumber\\  Olivier committed Apr 03, 2015 60  \begin{array}{c} {\scriptstyle \text{the rate of change}} \\ {\scriptstyle \text{of $B$ for the system}} \end{array} & = & \begin{array}{c} {\scriptstyle \text{the rate of change}} \\ {\scriptstyle \text{of $B$ within the}} \\ {\scriptstyle \text{control volume}} \end{array} & + & \begin{array}{c} {\scriptstyle \text{the net flow of $B$}} \\ {\scriptstyle \text{through the boundaries}} \\ {\scriptstyle \text{of the control volume}} \end{array} \nonumber  Olivier committed Apr 25, 2016 61  \end{IEEEeqnarray}  Olivier committed Apr 03, 2015 62   Olivier committed Mar 24, 2016 63  Since $B$ may not be uniformly distributed within the control volume, we like to express the term $\timederivative{B_\cv}$ as the integral of the volume density $\frac{B}{\vol}$ with respect to volume:  Olivier committed Apr 03, 2015 64  \begin{IEEEeqnarray}{rCl}  Olivier committed Apr 25, 2016 65  \timederivative{B_\cv} = \timederivative{} \iiint_\cv \frac{B}{\vol} \diff \vol = \timederivative{} \iiint_\cv \rho b \diff \vol \label{eq_secondbit}  Olivier committed Apr 03, 2015 66  \end{IEEEeqnarray}  Olivier committed Apr 23, 2017 67  Obtaining a value for this integral may be difficult, especially if the volume of the control volume CV is itself a function of time.  Olivier committed Apr 03, 2015 68   Olivier committed Apr 25, 2016 69  The term $\dot B_\net$ can be evaluated by quantifying, for each area element $\diff A$ of the control volume’s surface, the surface flow rate $\rho b V_\perp$ of property $B$ that flows through it (\cref{fig_cv_da}). The integral over the entire control volume surface CS of this term is:  Olivier committed Apr 03, 2015 70  \begin{IEEEeqnarray}{rCl}  Olivier committed Apr 25, 2016 71  \dot B_\net = \iint_\cs \rho b V_\perp \diff A = \iint_\cs \rho b \ (\vec V_\rel \cdot \vec n) \diff A \label{eq_thirdbit}  Olivier committed Apr 03, 2015 72  \end{IEEEeqnarray}  Olivier committed Mar 24, 2016 73  \begin{equationterms}  Olivier committed Apr 23, 2017 74 75 76 77 78 79  \item where flows and velocities are positive outwards and negative inwards by convention, \item \tab CV \tab is the control volume, \item \tab CS \tab\tab is the the control surface (enclosing the control volume), \item \tab $\vec n$ \tab\tab\tab is a unit vector on each surface element $\diff A$ pointing outwards, \item \tab $\vec V_\rel$ \tab is the local velocity of fluid relative to the control surface, \item and \tab $V_\perp \equiv \vec V_\rel \cdot \vec n$ is the local cross-surface speed.  Olivier committed Mar 24, 2016 80  \end{equationterms}  Olivier committed Apr 03, 2015 81 82 83  \begin{figure} \begin{center}  Olivier committed Mar 24, 2016 84  \includegraphics[width=0.7\textwidth]{concept_vrel_vecn.png}  Olivier committed Apr 03, 2015 85  \end{center}  Olivier committed Apr 23, 2017 86  \supercaption{Part of the system may be flowing through an arbitrary piece of the control surface with area $\diff A$. The $\vec n$ vector defines the orientation of $\diff A$ surface, and by convention is always pointed outwards.}{\wcfile{System control volume intregral analysis.svg}{Figure} \cczero \oc}  Olivier committed Apr 20, 2015 87  \label{fig_cv_da}  Olivier committed Apr 03, 2015 88 89  \end{figure}  Olivier committed Aug 16, 2015 90  By inserting equations~\ref{eq_secondbit} and~\ref{eq_thirdbit} into equation~\ref{eq_rtt_basic}, we obtain:  Olivier committed Apr 20, 2015 91  \begin{mdframed}\vspace{-0.5cm}\begin{IEEEeqnarray}{rCl}  Olivier committed Apr 25, 2016 92  \timederivative{B_\sys} & = & \timederivative{} \iiint_\cv \rho b \diff \vol + \iint_\cs \rho b \ (\vec V_\rel \cdot \vec n) \diff A \label{eq_rtt}  Olivier committed Apr 03, 2015 93  \end{IEEEeqnarray}  Olivier committed Apr 20, 2015 94  \end{mdframed}  Olivier committed Apr 03, 2015 95   Olivier committed Apr 25, 2016 96  Equation~\ref{eq_rtt} is named the \vocab{Reynolds’ transport theorem}; it stands now as a general, abstract accounting tool, but as we soon replace $B$ by meaningful variables, it will prove extremely useful, allowing us to quantify the \emph{net} effect of the flow of a system through a volume for which border properties are known.  Olivier committed Apr 03, 2015 97   Olivier committed May 11, 2015 98  In the following sections we are going to use this equation to assert four key physical principles (\S\ref{ch_conservation_equations}) in order to analyze the flow of fluids:  Olivier committed Apr 03, 2015 99 100  \begin{itemize} \item mass conservation;  Olivier committed Apr 23, 2017 101 102  \item change of linear momentum; \item change of angular momentum;  Olivier committed Apr 03, 2015 103 104 105  \item energy conservation. \end{itemize}  Olivier committed Mar 24, 2016 106 107  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  Olivier committed Mar 18, 2019 108 \section{Balance of mass}  Olivier committed Apr 03, 2015 109   Olivier committed Sep 23, 2017 110  \commonsvideo{Niccolò Paganini - Caprice No.5 - David Hernando.ogv}{https://frama.link/yVesHaSk}{with sufficient skills (and lots of practice!), it is possible for a musician to produce an uninterrupted stream of air into an instrument while still continuing to breathe, a technique called \vocab{circular breathing}. Can you identify the different terms of eq.~\ref{eq_rtt_mass} as they apply to the clarinetist’s mouth?}{David Hernando Vitores (\ccbysa)}  Olivier committed Apr 25, 2016 111  In this section, we focus on simply asserting that mass is conserved (eq.~\ref{eq_massconservation} p.\pageref{eq_massconservation}). Our study of the fluid’s properties at the borders of the control volume is made by replacing variable $B$ by mass $m$. Thus $\timederivative{B}$ becomes $\timederivative{m_\sys}$, which by definition is~zero.\\  Olivier committed Apr 03, 2015 112 113  In a similar fashion, $b \equiv B/m = m/m = 1$ and consequently the Reynolds transport theorem (\ref{eq_rtt}) becomes: \begin{IEEEeqnarray}{CCCCCCC}  Olivier committed Apr 20, 2015 114  \timederivative{m_\sys} & = & 0 & = & \timederivative{} \iiint_\cv \rho \diff \vol & + & \iint_\cs \rho \ (\vec V_\rel \cdot \vec n) \diff A \label{eq_rtt_mass}\\\nonumber\\  Olivier committed Apr 03, 2015 115 116 117  \begin{array}{c} {\scriptstyle \text{the time change}} \\ {\scriptstyle \text{of the system’s mass}} \end{array} & = & 0 & = & \begin{array}{c} {\scriptstyle \text{the rate of change}} \\ {\scriptstyle \text{of mass inside}} \\ {\scriptstyle \text{the control volume}} \end{array} & + & \begin{array}{c} {\scriptstyle \text{the net mass flow}} \\ {\scriptstyle \text{at the borders}} \\ {\scriptstyle \text{of the control volume}} \end{array} \nonumber \end{IEEEeqnarray}  Olivier committed Aug 12, 2015 118  This equation~\ref{eq_rtt_mass} is often called \vocab{continuity equation}. It allows us to compare the incoming and outgoing mass flows through the borders of the control volume.  Olivier committed Apr 03, 2015 119   Olivier committed Apr 23, 2017 120  When the control volume has well-defined inlets and outlets through which the term $\rho (\vec V_\rel \cdot \vec n)$ can be considered uniform (as for example in \cref{fig_cv_continuity_simple}), this equation reduces to:  Olivier committed Apr 03, 2015 121  \begin{IEEEeqnarray}{rCl}  Olivier committed Apr 25, 2016 122 123  0 & = & \timederivative{} \iiint_\cv \rho \diff \vol + \sum_\out \left\{ \rho V_\perp A\right\} + \sum_\inn \left\{ \rho V_\perp A\right\}\label{eq_rtt_mass_simple}\\ & = & \timederivative{} \iiint_\cv \rho \diff \vol + \sum_\out \left\{ \rho |V_\perp| A\right\} - \sum_\inn \left\{ \rho |V_\perp| A\right\} \nonumber\\  Olivier committed Apr 23, 2017 124  & = & \timederivative{} \iiint_\cv \rho \diff \vol + \sum_\out \left\{ \dot m \right\} - \sum_\inn \left\{ \dot m \right\}\label{eq_rtt_mass_simple_two}  Olivier committed Apr 03, 2015 125 126  \end{IEEEeqnarray}  Olivier committed Apr 25, 2016 127  In equation~\ref{eq_rtt_mass_simple}, the term $\rho V_\perp A$ at each inlet or outlet corresponds to the local mass flow~$\pm \dot m$ (positive inwards, negative outwards) through the boundary.  Olivier committed Apr 03, 2015 128 129 130  \begin{figure} \begin{center}  Olivier committed Mar 24, 2016 131  \includegraphics[width=0.8\textwidth]{simple_cv_fnet.png}  Olivier committed Apr 03, 2015 132  \end{center}  Olivier committed Apr 23, 2017 133  \supercaption{A control volume for which the system’s properties are uniform at each inlet and outlet. Here eq.~\ref{eq_rtt_mass} translates as $0 = \timederivative{} \iiint_\cv \rho \diff \vol + \rho_3 |V_{\perp 3}| A_3 + \rho_2 |V_{\perp 2}| A_2 - \rho_1 |V_{\perp 1}| A_1$.}{\wcfile{Integral analysis angular momentum sketch.svg}{Figure} \cczero \oc}  Olivier committed Apr 20, 2015 134  \label{fig_cv_continuity_simple}  Olivier committed Apr 03, 2015 135 136  \end{figure}  Olivier committed Apr 23, 2017 137 138  With equation~\ref{eq_rtt_mass_simple_two} we can see that when the flow is steady (\S\ref{ch_classification_of_fluid_flows}), the last two terms amount to zero, and the integral $\iiint_\cv \rho \diff \vol$ (the total amount of mass in the control volume) does not change with time.\dontbreakpage  Olivier committed Apr 03, 2015 139 140   Olivier committed Mar 24, 2016 141 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  Olivier committed Mar 18, 2019 142 \section{Balance of linear momentum}  Olivier committed Apr 03, 2015 143   Olivier committed Apr 23, 2017 144  In this section, we apply Newton’s second law: we assert that the variation of system’s linear momentum is equal to the net force being applied to it (eq.~\ref{eq_secondlaw} p.\pageref{eq_secondlaw}). Our study of the fluid’s properties at the control surface is carried out by replacing variable $B$ by the quantity $m \vec V$: momentum. Thus, $\timederivative{B_\sys}$ becomes $\timederivative{(m \vec V_\sys)}$, which is equal to $\vec F_\net$, the vector sum of forces applied on the system as it transits the control volume.\\  Olivier committed Aug 12, 2015 145  In a similar fashion, $b \equiv B/m = \vec V$ and equation~\ref{eq_rtt}, the Reynolds transport theorem, becomes:  Olivier committed Apr 03, 2015 146  \begin{IEEEeqnarray}{CCCCCCC}  Olivier committed Mar 27, 2019 147  \timederivative{(m \vec V_\text{sys})} & = & \vec F_\net & = & \timederivative{} \iiint_\cv \rho \vec V \diff \vol & + & \iint_\cs \rho \vec V \ (\vec V_\rel \cdot \vec n) \diff A \nonumber\\\label{eq_rtt_linearmom}\\  Olivier committed Apr 03, 2015 148 149 150  & & \begin{array}{c} {\scriptstyle \text{the vector sum of}} \\ {\scriptstyle \text{forces on the system}} \end{array} & = & \begin{array}{c} {\scriptstyle \text{the rate of change}} \\ {\scriptstyle \text{of linear momentum}} \\ {\scriptstyle \text{within the control volume}} \end{array} & + & \begin{array}{c} {\scriptstyle \text{the net flow of linear momen-}} \\ {\scriptstyle \text{tum through the boundaries}} \\ {\scriptstyle \text{of the control volume}} \end{array} \nonumber \end{IEEEeqnarray}  Olivier committed Apr 20, 2015 151  When the control volume has well-defined inlets and outlets through which the term $\rho \vec V (\vec V_\rel \cdot \vec n)$ can be considered uniform (\cref{fig_cv_continuity_simple}), this equation reduces to:  Olivier committed Apr 03, 2015 152  \begin{IEEEeqnarray}{rCl}  Olivier committed Apr 20, 2015 153  \vec F_\net & = & \timederivative{} \iiint_\cv \rho \vec V \diff \vol + \sum_\out \left\{ (\rho |V_\perp| A) \vec V\right\} - \sum_\inn \left\{ (\rho |V_\perp| A) \vec V\right\} \label{eq_rtt_linearmom_simple}\\  Olivier committed Apr 23, 2017 154  & = & \timederivative{} \iiint_\cv \rho \vec V \diff \vol + \sum_\out \left\{ \dot m \vec V\right\} - \sum_\inn \left\{ \dot m \vec V\right\}\label{eq_rtt_linearmom_simple_two}  Olivier committed Apr 03, 2015 155 156 157 158  \end{IEEEeqnarray} \begin{figure} \begin{center}  Olivier committed Mar 24, 2016 159  \includegraphics[width=0.8\textwidth]{simple_cv_fnet.png}  Olivier committed Apr 03, 2015 160  \end{center}  Olivier committed Apr 23, 2017 161  \supercaption{The same control volume as in \cref{fig_cv_continuity_simple}. Here, since the system’s properties are uniform at each inlet and outlet, eq.~\ref{eq_rtt_linearmom} translates as $\vec F_\net = \timederivative{} \iiint_\cv \rho \vec V \diff \vol + \rho_3 |V_{\perp 3}| A_3 \vec V_3 + \rho_2 |V_{\perp 2}| A_2 \vec V_2 - \rho_1 |V_{\perp 1}| A_1 \vec V_1$.}{\wcfile{Integral analysis angular momentum sketch.svg}{Figure} \cczero \oc}  Olivier committed Apr 20, 2015 162  \label{fig_cv_linearmom_simple}  Olivier committed Apr 03, 2015 163 164  \end{figure}  Olivier committed Apr 23, 2017 165  Let us observe the four terms of equation~\ref{eq_rtt_linearmom_simple_two} for a moment, for they are full of subtleties.  Olivier committed Apr 25, 2016 166   Olivier committed Apr 23, 2017 167  First, we notice that even if the flow is steady (an therefore that $\sum_\net \left(\dot m\right) = \SI{0}{\kilogram\per\second}$ since $\timederivative{} = 0$), the last two terms do not necessarily cancel each other (i.e. it is possible that $\sum_\net \left(\dot m\vec V\right) \neq \vec 0$).  Olivier committed Apr 25, 2016 168   Olivier committed Apr 23, 2017 169  The reverse also applies: it is quite possible that $\vec F_\net \neq \vec 0$ even if the net momentum flow through the boundaries is null (that is, even if $\sum_\net \left(\dot m \vec V\right) = \vec 0$). This would be the case, if $\iiint_\cv \rho \diff \vol$ (the total amount of momentum within the borders of the control volume) varies with time. Walking forwards and backwards within a rowboat would cause such an effect.  Olivier committed Apr 03, 2015 170   Olivier committed Sep 23, 2017 171  \youtubethumb{S6JKwzK37_8}{as a person walks, the deflection of the air passing around their body can be used to sustain the flight of a paper airplane (a \vocab{walkalong glider}). Can you figure out the momentum flow entering and leaving a control volume surrounding the glider?}{Y:sciencetoymaker (\styl)}  Olivier committed Apr 23, 2017 172  From equation~\ref{eq_rtt_linearmom_simple_two} therefore, we read that two distinct phenomena can result in a net force on the system:  Olivier committed Apr 03, 2015 173  \begin{itemize}  Olivier committed Apr 23, 2017 174 175  \item A difference between the values of $\dot m \vec V$ of the fluid at the entrance and exit of the control volume (caused, for example, by a deviation of the flow or by a mass flow imbalance); \item A change in time of the momentum $m \vec V$ within the control volume (for example, with the acceleration or the variation of the mass of the control volume).  Olivier committed Apr 03, 2015 176  \end{itemize}  Olivier committed Apr 23, 2017 177  These two factors may cancel each other, so that the system may well be able to travel through the control volume without any net force being applied to~it.  Olivier committed Mar 24, 2016 178 179   Olivier committed Apr 23, 2017 180 \clearpage %handmade  Olivier committed Mar 24, 2016 181 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  Olivier committed Mar 18, 2019 182 \section{Balance of angular momentum}  Olivier committed Apr 03, 2015 183   Olivier committed Sep 23, 2017 184  \youtubethumb{nmEe7Dq01AU}{pre-lecture briefing for this chapter, part 2/2}{\oc (\ccby)}  Olivier committed Apr 23, 2017 185  In this third spin on the Reynolds transport theorem, we assert that the change of the angular momentum of a system about a point X is equal to the net moment applied on the system about this point (eq.~\ref{eq_secondlawmom} p.\pageref{eq_secondlawmom}). Our study of the fluid’s properties at the borders of the control volume is made by replacing the variable $B$ by the angular momentum $\vec r_{\X m} \wedge m \vec V$. Thus, $\timederivative{B_\sys}$ becomes $\timederivative{(\vec r_{\X m} \wedge m \vec V_\sys)}$, which is equal to~$\vec M_\net$, the vector sum of moments applied on the system about point X as it transits through the control volume.  Olivier committed Apr 25, 2016 186   Olivier committed Aug 12, 2015 187  In a similar fashion, $b \equiv B/m = \vec r \wedge \vec V$ and equation~\ref{eq_rtt}, the Reynolds transport theorem, becomes:  Olivier committed Apr 03, 2015 188  \begin{IEEEeqnarray}{cCcCccc}  Olivier committed Apr 25, 2016 189  \timederivative{(\vec r_{\X m} \wedge m \vec V)_\sys} & = & \vec M_{\net, \X} & = & \timederivative{} \iiint_\cv \vec r_{\X m} \wedge \rho \vec V \diff \vol & + & \iint_\cs \vec r_{\X m} \wedge \rho \ (\vec V_\rel \cdot \vec n) \vec V \diff A \nonumber\\\label{eq_rtt_angularmom}\\  Olivier committed Apr 03, 2015 190 191  & & \begin{array}{c} {\scriptstyle \text{the sum of}} \\ {\scriptstyle \text{moments applied}} \\ {\scriptstyle \text{to the system}} \end{array} & = & \begin{array}{c} {\scriptstyle \text{the rate of change of}} \\ {\scriptstyle \text{the angular momentum}} \\ {\scriptstyle \text{in the control volume}} \end{array} & + & \begin{array}{c} {\scriptstyle \text{the net flow of angular}} \\ {\scriptstyle \text{momentum through the}} \\ {\scriptstyle \text{control volume’s boundaries}} \end{array} \nonumber \end{IEEEeqnarray}  Olivier committed Mar 24, 2016 192 193 194  \begin{equationterms} \item in which $\vec r_{\X m}$ is a vector giving the position of any mass $m$ relative to point $X$. \end{equationterms}  Olivier committed Apr 03, 2015 195   Olivier committed Sep 23, 2017 196  \youtubethumb{4cvGGxTsQx0}{rocket landing gone wrong. Can you compute the moment exerted by the top thruster around the base of the rocket as it (unsuccessfully) attempts to compensate for the collapsed landing leg?}{Y:SciNews (\styl)}  Olivier committed Mar 24, 2016 197  When the control volume has well-defined inlets and outlets through which the term $\vec r_{\X m} \wedge \rho \ (\vec V_\rel \cdot \vec n) \vec V$ can be considered uniform (\cref{fig_cv_angularmomentum_simple}), this equation reduces~to:  Olivier committed Apr 03, 2015 198  \begin{IEEEeqnarray}{rCCCCCl}  Olivier committed Apr 20, 2015 199  \vec M_{\net, \X} & = & \timederivative{} \iiint_\cv \vec r_{\X m} \wedge \rho \vec V \diff \vol &+& \sum_\out \left\{ \vec r_{\X m} \wedge \dot m \vec V\right\} - \sum_\inn \left\{ \vec r_{\X m} \wedge \dot m \vec V\right\} \nonumber\\\label{eq_rtt_angularmom_simple}  Olivier committed Apr 03, 2015 200 201  \end{IEEEeqnarray}  Olivier committed Apr 25, 2016 202  \begin{figure}[ht!]%handmade  Olivier committed Apr 03, 2015 203  \begin{center}  Olivier committed Mar 24, 2016 204  \includegraphics[width=0.6\textwidth]{simple_cv_mnet.png}  Olivier committed Apr 03, 2015 205  \end{center}  Olivier committed Sep 18, 2015 206  \supercaption{A control volume for which the properties of the system are uniform at each inlet or outlet. Here the moment about point X is $\vec M_{\net, \X} \approx \timederivative{} \iiint_\cv \vec r_{\X m} \wedge \rho \vec V \diff \vol + \vec r_{2} \wedge |\dot m_2| \vec V_2 - \vec r_{1} \wedge |\dot m_1| \vec V_1$.}{\wcfile{Integral analysis angular momentum sketch.svg}{Figure} \cczero \oc}  Olivier committed Apr 20, 2015 207  \label{fig_cv_angularmomentum_simple}  Olivier committed Apr 03, 2015 208 209  \end{figure}  Olivier committed Aug 12, 2015 210  Equation~\ref{eq_rtt_angularmom_simple} allows us to quantify, with relative ease, the moment exerted on a system based on inlet and outlet velocities of a control volume.  Olivier committed Apr 03, 2015 211 212   Olivier committed Apr 25, 2016 213 \dontbreakpage %handmade  Olivier committed Mar 24, 2016 214 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  Olivier committed Mar 18, 2019 215 \section{Balance of energy}  Olivier committed Apr 03, 2015 216   Olivier committed Apr 25, 2016 217  We conclude our frantic exploration of control volume analysis with the first principle of thermodynamics. We now simply assert that the change in the energy of a system can only be due to well-identified transfers (eq.~\ref{eq_firstprinciple} p.\pageref{eq_firstprinciple}). Our study of the fluid’s properties at the borders of the control volume is made by replacing variable $B$ by an amount of energy~$E_\sys$. Now, $\timederivative{E_\sys}$ can be attributed to three contributors:  Olivier committed Apr 03, 2015 218   Olivier committed Apr 25, 2016 219  \timederivative{E_\sys} = \dot Q_{\net \ \inn} + \dot W_\text{shaft, net in} + \dot W_\text{pressure, net in}  Olivier committed Apr 03, 2015 220 221  \begin{equationterms}  Olivier committed Apr 23, 2017 222 223 224  \item where \tab $\dot Q_{\net \ \inn}$ \tab is the net power transfered as heat; \item \tab $\dot W_\text{shaft, net in}$ \tab is the net power added as work with a shaft; \item and \tab $\dot W_\text{pressure, net in}$ \tab is the net power required to enter and leave the control volume.  Olivier committed Apr 03, 2015 225  \end{equationterms}  Olivier committed Apr 25, 2016 226  It follows that $b \equiv B/m = E/m \equiv e$; and $e$ is broken down into  Olivier committed Apr 03, 2015 227 228 229 230  e = i + e_k + e_p \begin{equationterms}  Olivier committed Apr 25, 2016 231 232 233  \item where \tab $i$ \tab\tab is the specific internal energy (\si{\joule\per\kilogram}); \item \tab $e_k$ \tab the specific kinetic energy (\si{\joule\per\kilogram}); \item and \tab $e_p$ \tab the specific potential energy (\si{\joule\per\kilogram}).  Olivier committed Apr 03, 2015 234 235  \end{equationterms}  Olivier committed Aug 12, 2015 236  Now, the Reynolds transport theorem (equation~\ref{eq_rtt}) becomes:  Olivier committed Apr 25, 2016 237  \begin{adjustwidth}{-3cm}{0cm}%handmade  Olivier committed Apr 03, 2015 238  \begin{IEEEeqnarray}{cCcCcCc}  Olivier committed Apr 25, 2016 239  \timederivative{E_\sys} & = & \dot Q_{\net \ \inn} + \dot W_\text{shaft, net in} + \dot W_\text{pressure, net in} & = & \timederivative{} \iiint_\cv \rho\ e \diff \vol & + & \iint_\cs \rho\ e \ (\vec V_\rel \cdot \vec n) \diff A \nonumber\\\label{eq_rtt_energy}  Olivier committed Apr 03, 2015 240  \end{IEEEeqnarray}  Olivier committed Apr 25, 2016 241  \end{adjustwidth}  Olivier committed Apr 03, 2015 242 243   Olivier committed Apr 03, 2015 244   Olivier committed May 01, 2016 245 246 247 248 249 250 251 252 253 254 255 256 257 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Limits of integral analysis} \label{ch_limits_integral_analysis} Integral analysis is an incredibly useful tool in fluid dynamics: in any given problem, it allows us to rapidly describe and calculate the main fluid phenomena at hand. The net force exerted on the fluid as it is deflected downwards by a helicopter, for example, can be calculated using just a loosely-drawn control volume and a single vector equation. As we progress through exercise sheet 3, however, the limits of this method slowly become apparent. They are twofold. \begin{itemize} \item First, we are confined to calculating the \emph{net} effect of fluid flow. The net force, for example, encompasses the integral effect of all forces —due to pressure, shear, and gravity— applied on the fluid as it transits through the control volume. Integral analysis gives us absolutely no way of distinguishing between those sub-components. In order to do that (for example, to calculate which part of a pump’s mechanical power is lost to internal viscous effects), we would need to look within the control volume. \item Second, all four of our equations in this chapter only work in one direction. The value $\diff B_\sys / \diff t$ of any finite integral cannot be used to find which function $\rho b V_\perp \diff A$ was integrated over the control surface to obtain it. For example, there are an \emph{infinite} number of velocity profiles which will result in a net force of \SI{-12}{\newton}. Knowing the net value of an integral, we cannot deduce the conditions which lead to it.\\ In practice, this is a major limitation on the use of integral analysis, because it confines us to working with large swaths of experimental data gathered at the borders of our control volumes. From the wake below the helicopter, we deduce the net force; but the net force tells us nothing about the shape of the wake. \end{itemize} Clearly, in order to overcome these limitations, we are going to need to open up the control volume, and look at the details of the flow within — perhaps by dividing it into a myriad of sub-control volumes. This is what we set ourselves to in chapter~4, with a thundering and formidable methodology we shall call \vocab{derivative analysis}. % ta-daaaa!  Olivier committed Apr 03, 2015 258   Olivier committed Apr 02, 2017 259 \atendofchapternotes